Mode I Fracture at Spot Welds in Dual-Phase Steel: An Application of Reverse Digital Image Correlation

Strain fields in 600 grade dual-phase steel V-notch tensile specimens, both with and without a spot weld, were measured after mode I fracture initiation. Starting with the final image of a fully developed crack, a novel reverse digital image correlation (DIC) analysis was used to determine the path that the crack followed at the onset of fracture as well as the crack tip deformation field. This gave the pixel coordinates of grid points on both sides (i.e. fracture surfaces) of the crack path in the undeformed image. Strain fields that develop in the base material regions surrounding the two fracture surfaces were subsequently measured with forward DIC analysis. Steady state values of the crack tip opening displacement (CTOD) and crack tip opening angle (CTOA), which are important fracture parameters, were measured for the base DP600 metal. Notch tip opening displacement (NTOD) and notch tip opening angle were also measured. It was found that steady state values of the CTOD and CTOA are reached within 2 mm or so of crack growth following completion of the flat-to-slant transition of the fracture surface and stabilization of the crack tunneling effect.     

Path-independent integral for the sharp V-notch in longitudinal shear problem

By applying Noether’s theorem to the elastic energy density in longitudinal shear problem, it is shown that its symmetry-transformations of material space can be expressed by the real and imaginary parts of an analytic function. This kind of the symmetry-transformations leads to the existence of a conservation law in material space, which does not belong to trivial conservation laws and whose divergence-free expression gives a path-independent integral. It is found that by adjusting the analytic function, a finite value can be obtained from this path-independent integral calculated around the material point with any order singularity. For a sharp V-notch placed on the edge of homogenous materials and/or the interface of bi-materials, application shows that the finite value obtained from this path-independent integral is directly related to the notch stress intensity factor (NSIF) and does not depend on the location of integral endpoints chosen respectively along two traction-free surfaces of which form a notch opening angle. Usability is presented in an example to estimate the NSIF of a bi-material plate.     

A Zener–Stroh crack interacting with a coated inclusion with generalized Irwin plastic zone correction

The elastic–plastic fracture behavior of a Zener–Stroh crack interacting with a coated inclusion in composite materials has been investigated with crack tip plastic zone corrections. With the distributed dislocation method, the crack problem is formulated into a set of singular integral equations which are solved numerically. The plastic zone sizes at the both crack tips are determined by a generalized Irwin model where Von Mises stress yielding criterion is used. The stress intensity factor (SIF), the plastic zone size (PZS), the crack tip opening displacement (CTOD) and the effective stress intensity factor have been evaluated. In the numerical examples, the influence of the inclusion shear modulus, the coating-layer thickness and shear modulus, as well as the distance between the crack and inclusion, on the SIF, the PZS and the CTOD are discussed in detail. Numerical examples show that increasing the shear modulus or the thickness of the coating phase, the influence of the inclusion on the normalized SIF and the normalized PZS will be shielded.     

Fatigue strength of welded joints based on local, semi-local and nominal approaches

Analysis based on the so-called “local approach” is made to estimate the fatigue strength of welded joints. Numerical analyses or strain gauges are employed for finding the stress and/or strain state in the vicinity of the weld toe. The notch stress intensity factor (NSIF) approach applied to fillet welded joints, as far as the opening angle between the weld and the main plate surface is constant (e.g. 135°, typical for many fillet welds), is able to rationalise the fatigue strength data both for different joint geometries and absolute dimensions. The NSIF approach has been previously developed as an extension of the Linear Elastic Fracture Mechanics (LEFM) to open V-notches and is based on the exponential local stress field around the V-notch tip. Several different “local approaches”, although simpler and more practical than the NSIF, are based on the stress (or strain) values determined beyond the exponential local one. To distinguish such approaches from the NSIF based one, we define the former as semi-local or nominal approaches while the latter is a local approach. The paper underlines that the local approaches, differently from the other ones, are able to unify in a single scatter band the fatigue strength data obtained from welded joints having different geometry and absolute dimensions.     

Pseudo plane stress plastic field for mode I crack growth

The pseudo plane stress field for a mode I crack growth is analyzed for both perfectly plastic and power law hardening plastic materials. When finite strain is taken into account, it is found that for perfectly plastic materials, the plastic domain is a narrow strip ahead of the crack tip. For power law hardening plastic material, the plastic domain contains a strip and a region ahead of the strip. The fracture criterion is discussed. The energy dissipated in the plastic strip is found to be proportional to the square of the thickness. Singular solutions to the field are ruled out by analysis.     

A new boundary element approach of modeling singular stress fields of plane V-notch problems

In this paper, a new boundary element (BE) approach is proposed to determine the singular stress field in plane V-notch structures. The method is based on an asymptotic expansion of the stresses in a small region around a notch tip and application of the conventional BE in the remaining region of the structure. The evaluation of stress singularities at a notch tip is transformed into an eigenvalue problem of ordinary differential equations that is solved by the interpolating matrix method in order to obtain singularity orders (degrees) and associated eigen-functions of the V-notch. The combination of the eigen-analysis for the small region and the conventional BE analysis for the remaining part of the structure results in both the singular stress field near the notch tip and the notch stress intensity factors (SIFs).Examples are given for V-notch plates made of isotropic materials. Comparisons and parametric studies on stresses and notch SIFs are carried out for various V-notch plates. The studies show that the new approach is accurate and effective in simulating singular stress fields in V-notch/crack structures.     

Analytical representation of the non-square-root singular stress field at a finite angle sharp notch

The stress field near the tip of a finite angle sharp notch is singular. However, unlike a crack, the order of the singularity at the notch tip is less than one-half. Under tensile loading, such a singularity is characterized by a generalized stress intensity factor which is analogous to the mode I stress intensity factor used in fracture mechanics, but which has order less than one-half. By using a cohesive zone model for a notional crack emanating from the notch tip, we relate the critical value of the generalized stress intensity factor to the fracture toughness. The results show that this relation depends not only on the notch angle, but also on the maximum stress of the cohesive zone model. As expected the dependence on that maximum stress vanishes as the notch angle approaches zero. The results of this analysis compare very well with a numerical (finite element) analysis in the literature. For mixed-mode loading the limits of applicability of using a mode I failure criterion are explored.     

A new version of the Neuber rule accounting for the influence of the notch opening angle for out-of-plane shear loads

The paper deals with nonlinear stress and strain distributions at the root of sharp and rounded notches with different opening angles under antiplane shear loading and small scale yielding. In order to make an easier comparison with the Neuber rule, the material is thought of as obeying the particular nonlinear law used in the past just by Neuber.By solving the linear differential equation resulting from the use of the hodograph transformation, a new relationship linking linear and nonlinear stress and strain concentrations is found. The relationship is written also in terms of the relevant notch stress intensity factors. In contrast with the Neuber rule, this relationship strictly depends on the notch opening angle. Even when the notch opening angle is zero, it does not match the Neuber Rule, but results in an additional factor 2 which is in agreement with Hult and McClintock’s solution when the notch tip radius tends to zero and the notch becomes a crack.     

The near tip stress and strain fields in cracked single crystals

The numerical analyses of stationary mathematically sharp Mode I crack in FCC and BCC crystals with elastic-ideally plastic (EIP) and fast hardening saturation (FHS) law are carried out in the present paper. From the calculated results, it is shown that: for the cases of small strain, EIP crystal cracks, the features of concentrated deformation patterns and the stress state in near-crack tip deformation fields are identical to the earlier analytical solutions, but along the angular sector boundaries, there exist narrow complex stress zones. The overall characteristics of deformation patterns for the cases of EIP and FHS are similar. The behaviours of crack tip opening can be characterized by crack-tip-opening-displacement (CTOD). For the case of FHS, finite deformation BCC crystal crack, our calculations are qualitatively in agreement with recent experimental observations. The project supported by National Natural Science Foundation of China     

Analysis of crack extension in anisotropic materials based on local normal stress

The normal stress ratio theory is applied to predict crack extension behavior in center-notched unidirectional graphite-epoxy of arbitrary fiber axis orientation, subjected to arbitrary far-field planar loading. The theory is applied within analytical solutions for two infinite plate geometries: a plate with a sharp center crack, and a plate with an elliptical center flaw. A critical analytical case is identified suggesting that application of the theory within a stress solution modelling crack tip shape may increase the accuracy of crack growth direction predictions. Crack extension direction, location of crack extension, and critical stress predictions of the theory are compared to those obtained from experiments on specimens subjected to tensile, shear, and mixed-mode far-field loading. The comparison shows that, applied within each analytical solution, the normal stress ratio theory provides verifiable predictions of crack growth behavior. By modelling actual notch tip shape, the elliptical notch solution is able to provide accurate qualitative predictions of the origin of crack extension along the periphery of a cut notch tip in a way that the sharp crack analysis cannot. The sharp notch solution appears to provide slightly more accurate crack growth direction predictions, however. Also, in predicting critical applied far-field stresses, the sharp crack solution appears to exhibit a stronger ability to model subtle experimental trends.     

基于奇异强度因子的V型切口应力场和N-SIF评估方法

根据线弹性断裂力学理论，V形切口处的应力场具有奇异性，应力值趋于无穷大，峰值应力不能直接用于评定疲劳强度。通过引入了奇异强度因子“as”，单边缺口应力分布和缺口应力强度因子(N-SIF)的半解析公式被推导。考虑张开角和几何尺寸等因素，基于奇异强度因子拟合得到了切口应力评估的简易公式，可用于切口应力场和N-SIF值的快速评估。将简易公式评估结果与有限元结果以及传统文献结果进行对比分析，结果表明，本文简易公式可以准确地预报拉伸载荷下单边V型切口角平分线上的应力场和N-SIF值，实现了切口试样应力场的快速评估。     

METHOD TO CALCULATE STRESS INTENSITY FACTOR OF V-NOTCH IN BI-MATERIALS

Based on Zak＇s stress function, the eigen-equation of stress singularity ofbi-materials with a V-notch was obtained. A new definition of stress intensity factor for a perpendicular interfacial V-notch of bi-material was put forward. The effects of shear modulus and Poisson＇s ratio of the matrix material and attaching material on eigen-values were analyzed. A generalized expression for calculating/（i of the perpendicular V-notch of bi-materials was obtained by means of stress extrapolation. Effects of notch depth, notch angle and Poisson＇s ratio of materials on the singular stress field near the tip of the V-notch were analyzed systematically with numerical simulations. As an example, a finite plate with double edge notches under uniaxial uniform tension was calculated by the method presented and the influence of the notch angle and Poisson＇s ratio on the stress singularity near the tip of notch was obtained.     

Fracture investigation at V-notch tip using coherent gradient sensing (CGS)

Local deformation field and fracture characterization of mode I V-notch tip are studied using coherent gradient sensing (CGS). First, the governing equations that relate to the CGS measurements and the elastic solution at mode I V-notch tip are derived in terms of the stress intensity factor, material constant, notch angle and fringe order. Then, a series of CGS fringe patterns of mode I V-notch are simulated, and the effects of the notch angle on the shape and size of CGS fringe pattern are analyzed. Finally, the local deformation field and fracture characterization of mode I V-notch tip with different V-notch angles are experimentally investigated using three-point-bending specimen via CGS method. The CGS interference fringe patterns obtained from experiments and simulations show a good agreement. The stress intensity factor obtained from CGS measurements shows a good agreement with finite element results under K-dominant assumption.     

扩展边界元法分析切口和裂纹结构应力场的准确性

在线弹性理论中,切口/裂纹结构尖端区域存在奇异应力场,数值方法不易求解。本文建立的扩展边界元法(XBEM)对围绕尖端区域位移函数采用自尖端径向距离 的渐近级数展开式表达,其级数项的幅值系数作为基本未知量,而外部区域采用常规边界元法离散方程。两者方程联立求解可获得切口和裂纹结构完整的位移和应力场。扩展边界元法具有半解析法特征,适用于一般的切口和裂纹结构应力场分析,其解可精细描述从尖端区域到整体结构区域的应力场。作者研制了扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明扩展边界元法求解切口和裂纹结构应力场的准确性和有效性。     

A generalized stress intensity factor to be applied to rounded V-shaped notches

In the presence of sharp (zero radius) V-shaped notches the notch stress intensity factors (N-SIFs) quantify the intensities of the asymptotic linear elastic stress distributions. They are proportional to the limit of the mode I or II stress components multiplied by the distance powered 1 − λ_i from the notch tip, λ_i being Williams’ eigenvalues. When the notch tip radius is different from zero, the definition is no longer valid from a theoretical point of view and the characteristic, singular, sharp-notch field diverges from the rounded-notch solution very next to the notch. Nevertheless, N-SIFs continue to be used as parameters governing fracture if the notch root radius is sufficiently small with respect to the notch depth.Taking advantage of a recent analytical formulation able to describe stress distributions ahead of rounded V-notches, the paper gives a generalized form for the notch stress intensity factors, in which not only the opening angle but also the tip radius dimension is explicitly involved. Such parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case.     

On the intensity of linear elastic high order singularities ahead of cracks and re-entrant corners

The paper deals with high order elastic singular terms at cracks and re-entrant corners (sharp V-notches), which are commonly omitted in linear elastic analyses by the argument that the strain energy and displacements in the near-tip region should be bounded. The present analysis proves that these terms are fully included in the elastic part of complete elastic–plastic stress and strain solutions.The intensities of high order singular terms are found to be linked to the linear elastic stress intensity factor and the extension of the plastic zone along the crack bisector line. The smaller the plastic radius, the smaller the intensities of high order singular terms are.A physical justification of the existence of high order singular terms is provided on the basis of the strain energy density distribution detected along the crack bisector line. Finally, the influence of the V-notch opening angle is made explicit, discussing also the relationship between the singularity orders and the solution of a Williams’ type sinusoidal eigen-equation.     

Influence of stress triaxiality on damage and crack tip opening displacement parameters for steels

The dependence of the void growth parameter on the local stress triaxiality and local effective plastic strain near the crack tip of ductile materials provides the motivation to seek for parameters that could rank the ductility of steels. Experimental data for AS 1405-180, AS 1204-350, HY-80 and C---Mn steels show that the crack tip opening displacement (CTOD) at initiation δ_c decreases with increasing crack tip stress triaxiality. This trend is confirmed by analysis. As the critical local effective plastic strain ε_ec also decreases with increasing local stress triaxiality, the ratio δ_c/ε_ec is found to remain nearly constant or independent of the local constraint, i.e., the stress triaxiality. These parameters are given for a class of steels in this paper. Their association with the resistance to ductile fracture remains to be investigated.     

角度非均匀材料平面V形切口应力奇性分析

提出了一种确定角度非均匀材料平面V形切口尖端应力奇性指数的有效方法。首先,在弹性力学基本方程中引入V形切口尖端位移场的级数渐近展开,建立以位移为特征函数的变系数和非线性微分方程组。然后,采用微分求积法(DQM)求解微分方程组,可得到多阶应力奇性指数及其相对应的特征函数,该法具有公式简单、编程方便、计算量少和精度高等优点,可处理任意开口角度和任意材料组合的V形切口。典型算例验证了微分求积法的有效性和精确性。     

Effect of crack-tip shape on the near-tip field in glassy polymer

The physical nature of a crack tip is not absolutely sharp but blunt with finite curvature. In this paper, the effects of crack-tip shape on the stress and deformation fields ahead of blunted cracks in glassy polymers are numerically investigated under Mode I loading and small scale yielding conditions. An elastic–viscoplastic constitutive model accounting for the strain softening upon yield and then the subsequently strain hardening is adopted and two typical glassy polymers, one with strain hardening and the other with strain softening–rehardening are considered in analysis. It is shown that the profile of crack tip has obvious effect on the near-tip plastic field. The size of near-tip plastic zone reduces with the increase of curvature radius of crack tip, while the plastic strain rate and the stresses near crack tip enhance obviously for two typical polymers. Also, the plastic energy dissipation behavior near cracks with different curvatures is discussed for both materials.     

Higher order asymptotic fields at the tip of a sharp V-notch in a power-hardening material

The higher order asymptotic fields at the tip of a sharp V-notch in a power-hardening material for plane strain problem of Mode I are derived. The order hierarchy in powers ofr for various hardening exponentsn and notch angles β is obtained. The angular distributions of stress for several cases are plotted. The self-similarity behavior between the higher order terms is noticed. It is found that the terms with higher order can be neglected for the V-notch angle β>45°. Project supported by the National Natural Science Foundation of China (Nos. 10132010 and 10072033).